296 MR. W. H. L. RUSSELL ON THE 



XXI. — On the Solution of the Differential Resolvent. By 

 W. H. L. Russell, A.B. Communicated by the Rev. 

 Robert Harley, F.R.A.S. 



Read March nth, 1863. 



In the course of last year I was induced,, at the request of 

 Mr. Harley, to consider the very interesting differential 

 equation which he has denominated the " Differential 

 Resolvent.^^ I obtained the solution of the quartic re- 

 solvent by series which I summed by means of a triple 

 integral. But Professor Boole intimated that he had dis- 

 covered a process of transformation by wljich the quartic 

 could be solved by a single integral. This led me to 

 examine my own series, and I found that the series repre- 

 senting the solution of the quartic could be summed 

 by means of a single integral. I have since discovered 

 that the general resolvent can be solved by means of a 

 single integral. To effect this is the object of the present 

 paper. 



As the investigations are complicated, I shall first, to fix 

 the ideas, give the solution of the quartic. 



The quartic resolvent is the equation 



(Mfcifcj.),.. „ 



^ D(D-i)(D-2) y~ ' 



where e^=x. 



I have already expressed the solution of this equation in 

 series, which will be found in the ^ Proceedings' of the Man- 

 chester Society. One of these series is the following : — 



^ , 4 ' 4 ' 4 . , V4 ' 4/V4 • 4/W '4/ , .6 , 

 '"^ 5 •4- 3^ (8-5) • (7.4) . (6.3) ^- 



